The inherent asymmetry of the electric transport in graphene is attributed to Klein tunneling across barriers defined by $textit{pn}$-interfaces between positively and negatively charged regions. By combining conductance and shot noise experiments we determine the main characteristics of the tunneling barrier (height and slope) in a high-quality suspended sample with Au/Cr/Au contacts. We observe an asymmetric resistance $R_{textrm{odd}}=100-70$ $Omega$ across the Dirac point of the suspended graphene at carrier density $|n_{rm G}|=0.3-4 cdot 10^{11}$ cm$^{-2}$, while the Fano factor displays a non-monotonic asymmetry in the range $F_{textrm{odd}} sim 0.03 - 0.1$. Our findings agree with analytical calculations based on the Dirac equation with a trapezoidal barrier. Comparison between the model and the data yields the barrier height for tunneling, an estimate of the thickness of the $textit{pn}$-interface $d < 20$ nm, and the contact region doping corresponding to a Fermi level offset of $sim - 18$ meV. The strength of pinning of the Fermi level under the metallic contact is characterized in terms of the contact capacitance $C_c=19 times 10^{-6}$ F/cm$^2$. Additionally, we show that the gate voltage corresponding to the Dirac point is given by the work function difference between the backgate material and graphene.
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